Papers

We present the algebraic structure of bosonic quantum error-correcting codes on genus-two Riemann surfaces. We explicitly construct the code words as automorphic forms and analytically generate the full tower of code spaces at all weights. We prove a fundamental no-go theorem: for any genus greater than one, the stabilizer group is non-amenable, forcing a strictly positive spectral gap in the stabilizer Hamiltonian. Consequently, no normalizable quantum state can satisfy all stabilizer conditions. This sharply contrasts with standard Gottesman-Kitaev-Preskill (GKP) codes, where the amenability of the stabilizer group $\mathbb{Z}^2$ permits approximate code words with arbitrary precision.

The chiral limit of the Bistritzer–MacDonald model for twisted bilayer graphene has exactly flat bands at magic angles. We show that exact flatness can persist beyond the chiral limit with nonchiral tunnelling potentials in the usual symmetry class. We construct a family of nonchiral tunnelling potentials for which the Hamiltonian has at least two zero-energy states at every Bloch momentum and for every real value of the nonchiral coupling. This gives a family of counterexamples to Open Problem~3 of Zworski's survey \cite{ZworskiSurvey}. Conversely, we prove that every admissible potential with this property belongs to this family, thereby obtaining a complete characterization. By contrast, we show that the standard BM model, namely, BM Hamiltonians with first harmonic tunnelling potentials, does not admit exactly flat bands at the magic angle except possibly at a discrete set of nonchiral coupling strengths, with no finite accumulation point. To prove this, we introduce an explicit invertibility criterion. The criterion requires the projected perturbation to have a nonzero scalar coefficient at some Bloch momentum, obstructing exact flatness for general nonchiral tunnelling. A local Taylor expansion argument verifies the criterion for the standard BM potentials. For the family constructed above, however, the projected perturbation vanishes at every Bloch momentum, so the invertibility criterion fails.

In this paper, we initiate the study of colored exponential random graph models (ERGMs), a class of exponential-family models for networks with multiple types of edge relations. Using the framework of probability graphons, we first derive a variational representation for the limiting free energy, whose maximizers determine the asymptotic structure of typical samples from the model. Then we identify several general families of colored ERGMs exhibiting replica symmetry, where the variational problem has constant maximizers and the model asymptotically concentrates on product colorings with independent edges. For general colored ERGMs, we derive Euler-Lagrange fixed-point equations for the variational maximizers, which in turn yield a general high-temperature uniqueness criterion. In the complementary zero-temperature regime, we establish a two-level selection principle: the leading energy term determines the ground states, while the lower-order energy terms, combined with entropy, act as a tie-breaker to determine the asymptotic zero-temperature structure of the model. We illustrate this principle through the induced wedge and rainbow triangle ERGMs. Both models have natural interpretations in multitype networks, and their zero-temperature limits exhibit interesting structures that connect to well-known results in extremal combinatorics. We further establish finite-temperature symmetry breaking for both these models and complement the rigorous results with numerical experiments.

We study the existence of gravitational radiation at infinity for pure-radiation Robinson-Trautman metrics with one spacelike Killing vector and a cosmological constant $\Lambda$ of arbitrary sign. These metrics describe "photon rockets" with point, string, and sheet sources moving through spacetime while emitting null radiation. For $\Lambda\neq 0$, we use criteria for the existence of gravitational radiation based on the asymptotic super-Poynting vector. For point sources, we extend to $\Lambda\neq 0$ the known $\Lambda=0$ result that the Kinnersley rocket is the only rocket without gravitational radiation. However, for string and sheet sources, new possibilities without gravitational radiation arise. For $\Lambda>0$, we show that the vanishing of the canonical asymptotic super-Poynting vector—computed with respect to the unit normal to scri—exactly characterizes the absence of radiation. On the other hand, for $\Lambda<0$, we show that the appropriate criterion is the vanishing of the components normal to scri of the asymptotic super-Poynting vectors associated with any unit timelike vector tangent to scri or, equivalently, the proportionality between the Cotton-York tensor and the holographic stress tensor at scri. In particular, we provide explicit examples of metrics for which the Cotton-York and holographic stress tensors commute, yet gravitational radiation is present because the two tensors are not proportional. We also determine the principal null directions (PNDs) of the rescaled Weyl tensor at scri for both signs of $\Lambda$ and relate their geometry relative to the normal to scri to the tensorial criteria for gravitational radiation.

The work introduces the novel Dunkl two-photon and Dunkl-Schrödinger algebras as reflection-extended counterparts of the standard six-dimensional two-photon and Schrödinger algebras. These extensions are realized through the presence of an involutive generator representing a reflection, and they reduce to the undeformed two-photon and Schrödinger algebras in the absence of this additional generator in an appropriate limit. For both algebraic structures, which are shown to be isomorphic, we derive the associated cubic polynomial Casimir invariant. Furthermore, we identify several structurally significant subalgebras and compute their corresponding quadratic Casimir invariants. Representations of the Dunkl two-photon algebra are constructed on the Fock space of the one-dimensional Dunkl harmonic oscillator and subsequently translated into holomorphic realizations within the associated Dunkl-Fock-Bargmann representation. Finally, for the Dunkl-Schrödinger algebra, we present an explicit vector field realization in terms of the Dunkl spatial derivative, while keeping time continuous. When applied to the Casimir invariants of the aforementioned subalgebras, this realization yields a family of six nontrivial Dunkl differential-difference equations in $1+1$ dimensions, including, in particular, a Dunkl version of the well-known heat-Schrödinger equation.

Fushchych and Shtelen defined approximate symmetry by expanding the solution of a perturbed differential equation in the small parameter, replacing the original equation by a triangular system for the expansion coefficients. We extend this construction to arbitrary order. A coefficient-map formulation yields the order-$p$ Fushchych–Shtelen (FS) system and establishes, under explicit hypotheses, a correspondence with the Baikov–Gazizov–Ibragimov (BGI) method. The two frameworks are compared for a perturbed cubic wave equation. Up to equivalences, we classify the nonlinearities admitting an FS dilation through second order. The classification consists of a generic power family and logarithmic branches at exceptional exponents. Among perturbations nontrivial at first order, vertical BGI continuations select a proper subfamily of one FS branch, and a second common branch arises when the perturbation first enters at second order; the two methods are related but not interchangeable. Joint reduction by the Lorentz algebra and the inherited FS dilation integrates all resulting FS normal forms; the explicit solutions of the 1989 Fushchych–Shtelen letter are recovered as the first-order members of this reduction. In the part common to both frameworks, the logarithmic corrections arise from the expansion of a power law with a perturbation-dependent exponent. Finally, periodic travelling waves are used to examine the long-scale validity of the FS expansion. The first correction is obtained by quadrature, the second-order secular structure is isolated, and phase renormalization produces bounded order-consistent approximations. For integer-power members of the generic family, the amplitude dependence of the corrected wavenumber agrees with the scaling weights obtained from the symmetry classification.

We derive state-dependent bounds on the difference between two quantum evolutions generated by unbounded Hamiltonians sharing a common form domain. The main technical tool is a second integration by parts, performed at the level of sesquilinear forms rather than at the operator level, which removes the need for a common invariant operator domain. The resulting estimate involves the norm of the time-integrated difference of the two generators, rather than the integral of its norm, and is therefore sensitive to the averaging effects produced by fast-oscillating terms. As an application we prove a quantitative bound on the rotating-wave approximation for spin-boson models with a structured boson field, described by an arbitrary massive dispersion relation on a general measure space and by a suitable class of form factors. The proof involves a careful analysis of the high-frequency scaling. The bound holds on a dense subspace of states, is fully explicit, and all the constants entering it depend only on the parameters of the model and not on the frequency scale, so that the approximation becomes exact in the limit of large frequency.

The generalized Bethe ansatz framework formulated in [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)] provides a unified framework to find exact solutions to quantum many-body systems with time-dependent coupling strengths with periodic boundary conditions. In this work we extend this framework to the case of open boundary conditions where in addition to the time-dependent interactions in the bulk, the boundary conditions are explicitly time-dependent. We show that for integrable time-dependent bulk coupling strengths, the generalized Bethe ansatz framework provides the time-dependent boundary conditions compatible with integrability and reduces the time-dependent Schrodinger equation to a set of matrix difference equations called the boundary quantum Knizhnik-Zamolodchikov (BqKZ) equations. The solution to the BqKZ equations provides the explicit form of the exact wavefunction. We further show that the RG invariants of the corresponding static model identify with the dynamical invariants in the time-dependent model.

We prove a correspondence between the Polchinski flow and the Connes-Kreimer approaches to perturbative renormalisation, in the sense that the first yields the same renormalisation as the latter. More precisely, we show that an ansatz based on decorated graphs (Feynman diagrams),with a combinatorial renormalisation procedure solves the Polchinski equation. This result holds for very general Euclidean quantum field theories. We are able to derive from this the form of the renormalised potential which is, to the best of our knowledge, the most general one in the literature. The main ingredients of the proof are multiple morphism properties with respect to the renormalisation, as well as a novel duality formula for forests of decorated graphs.

We prove that a complete ALF Ricci-flat 4-manifold carrying a bounded Killing vector field is linearly stable if and only if it is locally hyperkähler. This applies uniformly to all known examples and in particular proves the instability of all of the metrics recently found by Li-Sun. The argument is based on infinitesimal Einstein-Maxwell deformations of the Ricci-flat metric associated with the Killing field. A separate simpler identity proves the instability of nonflat static Ricci-flat 4-metrics which include the static axisymmetric smooth Riemannian Myers/Korotkin-Nicolai metrics. To our knowledge, this proves the equivalence between linear stability and special holonomy and (anti-)selfduality for all known families of Ricci-flat 4-manifolds. In higher dimensions, for generalizations of AF manifolds, we show that stability forces the universal cover to split a line. This gives explicit destabilizing tensors on the Riemannian Myers-Perry instantons and the Riemannian Schwarzschild-Tangherlini metrics in all dimensions.

We reformulate the relaxation problem in statistical mechanics by making explicit what are the operational objects subject to relaxation: the local time statistics of the recorded signal $Z(t)$. These local time statistics are simply the estimated histograms of observations $\{Z(t_i)\}_{i=1}^M$ performed at uniformly random times $\{t_i\}_{i=1}^M$ by a clockless observer. The subject of prediction is a belief about a future fresh out-of-sample reading of a measurement outcome whose distribution is inferred from the mathematical model believed to be true. For finite bounded systems of $N\ge 1$ degrees of freedom global irreversible relaxation of predictions can occur but special initial conditions exist. The form of the predictions depends on certain loss functions whose choice is up to the particular observer. Finally, entropy is given a learning interpretation as mutual information between the observer and the unknown past of the system under consideration and, in complete generality, its stationary value depends on the information available.

Local constraints govern the low-energy physics of frustrated matter, but familiar ice-type rules constrain flux-like quantities and are insensitive to handedness. Here we show that handedness itself can be imposed as an exact local quantum constraint without selecting an axis in spin space. We construct positive-semidefinite, SU(2)-invariant parent Hamiltonians whose complete zero-energy space on a tetrahedron has a prescribed chirality sign, rather than selecting a particular chiral wave function. For spin-1/2 the local term is a rank-one projector onto a chiral tetrahedral singlet, while for arbitrary spin it factorizes as $B^\dagger B$ through a singlet-annihilation operator, with a completely characterized kernel given by the span of the globally rotated chiral color-ice states. For coherent states, the same zero-energy condition becomes an $S$-independent nonlinear constraint in which three spin directions determine the fourth through a Möbius transformation; compositions of these maps define constraint holonomies on extended lattices. Connecting the same local constraint in different ways produces qualitatively different collective regimes: corner-sharing lattices retain exponentially large quantum ground-state kernels, with rigorous lower bounds exceeding conventional ice benchmarks; edge-sharing lattices support subdimensional plane or line zero modes; while triangular constructions suppress nonuniform coherent deformations and contain the complete Anderson tower of tetrahedral magnetic order at exactly zero energy. Two inequivalent triangular coverings further show that harmonic zero-mode counting does not determine the size of the quantum kernel. These results establish a tractable setting in which local handedness, nonlinear constraint geometry, and quantum degeneracy can be disentangled and related directly to the connectivity of the constraint network.

By analyzing a stochastic Lotka-Volterra (LV) equation, we show that the underlying logic behind the diversity-stability debate in ecology can be framed as a dimensionality-induced critical phase transition, that is, a transition separating the regime dominated by statistical averaging and a systemic tipping point. This phase transition framework unifies the opposing ecological predictions in the diversity-stability debate and sets an intrinsic diversity ceiling for stochastic ecological communities.

We consider the massive Dirac operator (with positive mass) in the plane with an attractive Lorentz-scalar $\delta$-shell interaction of strength $\tau\in(-\infty,0)\setminus\{-2\}$ supported on a $C^\infty$-smooth curve $\Sigma\subset\mathbb{R}^2$ being a local deformation of the broken line. This singular interaction is defined by imposing a suitable transmission condition on the curve $\Sigma$ in the operator domain. Such a Dirac operator is self-adjoint and has a gap in the essential spectrum, whose size is explicit and depends on the mass and the interaction strength. We show that the number of discrete eigenvalues in the gap is finite. Under the assumption that one of the domains bounded by $\Sigma$ is convex, we prove that the corresponding Dirac operator has a non-empty discrete spectrum, provided that $\tau$ is either sufficiently small or sufficiently large in absolute value. The result holds for any perturbation of a broken line of any opening angle as described above, and this discrete spectrum is induced by the geometry, since for the same type of a singular interaction supported on the straight line the discrete spectrum is empty.

We establish sharp endpoint mapping properties for the wave operators $W_\pm(H,-\Delta)$ of two-dimensional Schrödinger operators $H=-\Delta+V$ with real-valued decaying potentials $V$. Together with the known non-endpoint $L^p$ theory, our results give a complete classification of the $L^p$ mapping properties of the two-dimensional wave operators, and reveal an unexpected reversal of the usual threshold paradigm at the endpoints $p=1$ and $p=\infty$. When zero is a regular point of $H$, the wave operators fail to be bounded on $L^1(\mathbb{R}^2)$ and on $L^\infty(\mathbb{R}^2)$, but they satisfy the atural substitute estimates of Calderón–Zygmund type: $$ L^1(\mathbb{R}^2)\longrightarrow L^{1,\infty}(\mathbb{R}^2),\ \ \ \mathcal{H}^1(\mathbb{R}^2)\longrightarrow L^1(\mathbb{R}^2),\ \ \ L^\infty(\mathbb{R}^2)\longrightarrow \mathrm{BMO}(\mathbb{R}^2). $$ When zero is instead a threshold singularity of the first kind—an s-wave resonance with no other threshold obstruction, the wave operators are bounded on both endpoint spaces $L^1(\mathbb{R}^2)$ and $L^\infty(\mathbb{R}^2)$. Thus, in dimension two, an s-wave resonance improves the endpoint behavior of the wave operators, in sharp contrast with dimensions $n\ge3$, where the only regular case is the favorable one. We also determine the endpoint behavior in the remaining zero-energy spectral configurations of $H$. A p-wave resonance obstructs both the $L^1$- and the $L^\infty$-boundedness of the wave operators, while in the zero-eigenvalue case we obtain necessary and sufficient conditions for endpoint boundedness, expressed in terms of the presence of s- and p-wave resonances and of explicit second-order harmonic moment cancellations satisfied by the zero-energy eigenfunctions.

We extend the functional determinant approach (FDA), previously restricted to static impurities, to the case of finite-mass impurities by using the mass-gap description of Fermi polarons [Phys. Rev. Lett. 135, 193401 (2025)]. The quadratic structure of the mass-gap model enables the exact evaluation of many-body spectra and Ramsey dynamics for mobile impurities. We show that this mass-gap FDA smoothly interpolates between the Fermi-edge singularity for infinitely heavy impurities and the emergence of quasiparticle weight for finite impurity mass. Our results demonstrate that the mass-gap FDA provides a transparent and computationally efficient framework to describe the quantum dynamics of mobile impurities.

We study the algebraic and differential structure of the Riemann curvature tensor of the higher-dimensional Kerr-NUT-(A)dS spaces, motivated by the eventual goal of finding an IDEAL (Intrinsic, Deductive, Explicit and ALgorithmic) characterization of this family of metrics. The special geometry of these spaces is governed by the principal tensor $h_{ab}$, a non-degenerate closed conformal Killing-Yano $2$-form, whose existence singles out the Kerr-NUT-(A)dS family; we therefore focus on the algebraic relationship between the Riemann curvature $R_{abcd}$ and $h_{ab}$. In our investigations we encountered an obstruction, which results in a no-go theorem: no non-trivial $2$-form, including $h_{ab}$ itself, can be constructed covariantly from the undifferentiated Riemann tensor alone. Despite that, we do characterize the family of Riemann-symmetric tensors that could be the curvature of a Kerr-NUT-(A)dS metric by an algebraic and a differential condition, stemming from the integrability of the defining equation of $h_{ab}$ and the second Bianchi identity. Our calculations apply in all higher dimensions, which becomes feasible by a judicious application of representation theoretic techniques related to a semi-direct product group $U(1)^n\rtimes S_n$ that stabilizes $h_{ab}$.

Fix the spectrum of a bipartite density matrix and randomize its eigenbasis according to Haar measure. We study the probability distributions induced on the spectra of the two marginal states. For arbitrary subsystem dimensions $m$ and $n$, the joint characteristic function of the reduced density matrices is expressed as a Harish-Chandra-Itzykson-Zuber integral whose external eigenvalues are the pairwise sums $x_i + y_j$. Repeated external eigenvalues are handled by confluent determinant limits. In the two-qubit case, we derive an explicit alternating-spline formula for the joint density of the two marginal Bloch radii. Its support is the Bravyi-Klyachko compatibility region. We also obtain a compact truncated-power formula for the Bloch-radius density of either individual qubit marginal. In the qubit-qutrit case, we derive a truncated-power formula for the qubit Bloch-radius density and a bivariate spline formula for the joint density of the largest and smallest eigenvalues of the qutrit marginal. The latter two variables determine the full qutrit spectrum because the trace is fixed. The derivations combine confluent HCIZ integrals, distributional Fourier inversion, orbital measures, and the SU(2) and SU(3) derivative principles. The resulting densities are piece-wise polynomial on chambers determined by subset sums of the fixed global eigenvalues, in agreement with the Duistermaat-Heckman description of projected coadjoint-orbit measures.

We study the stabilizability of a N -level quantum systems using an indirect control ar- chitecture mediated by a noisy ancilla. After outlining the general theoretical framework for arbitrary N , we focus on the two-level case (N = 2), for which we derive a complete set of necessary and sufficient conditions for asymptotic stability. Finally, we illustrate the analyti- cal results with a simple numerical example on a two-spin system, showing convergence to a Bell state for a set of random initial conditions.

In 2004, Khoroshkin proved that the extremal projector is equivalent to the dynamical twist. The Arnaudon-Buffenoir-Ragoucy-Roche (ABRR) equation, satisfied by the dynamical twist, yields a recursive formula for the coefficients in the extremal projector. The resulting summation formula is uniform and powerful in its applications to representation theory and Mickelsson-Zhelobenko reduction algebras. In this paper we generalize this recursive formula to the relative extremal projector, introduced by Conley and Sepanski in 2003. When the Levi subalgebra is the Cartan subalgebra, we recover the usual ABRR recursion. We also find a compact and explicit expression for the solution to the recursion. Our setting includes infinite-dimensional contragredient Lie superalgebras, Kac-Moody algebras, basic classical Lie superalgebras, and finite-dimensional reductive Lie algebras.

For periodic Schrödinger operators $H_V=-\Delta+V$ with bounded real-valued potentials on $\mathbb R^d$ with $d\ge2$, we show that for generic potentials, each endpoint of every spectral gap is attained by a single Bloch band at only finitely many quasimomenta, and has a nondegenerate Hessian at every attaining point. This proves the Spectral Edge Conjecture for periodic Schrödinger operators.

We establish the existence of smooth solutions to the symmetric maximal surface equation with degenerate boundary conditions. Moreover, we prove that these solutions maximize the associated area functionals. This result serves as the Lorentzian analogue of minimal graphs in hyperbolic spaces together with their associated area minimizing problem.

This paper systematically investigates admissible configurations of the interaction between a rarefaction wave (RW) and a dispersive shock wave (DSW) by constructing a two-step piecewise constant initial condition in the Gerdjikov–Ivanov (GI) equation with non-convex hydrodynamics. Owing to the two-valued mapping associated with the Riemann invariants, the GI equation also supports contact DSWs (CDSWs), which are included in the interaction analysis. Using Whitham modulation theory, we elucidate the self-similar descriptions of the incoming waves, derive the boundary matching conditions, and obtain analytical expressions of the full evolution before, during, and after the interaction through generalized hodograph transformation. The results show that the interaction of a RW with either a DSW or a CDSW can exhibit two distinct scenarios: in one case, the two waves exchange spectral parameters after the interaction and fully separate, propagating independently with modified phases; in the other case, the wave structures merge upon interaction and subsequently evolve into either a soliton train or a small-amplitude harmonic wave for a classical DSW, and into an algebraic soliton train for a CDSW. The theoretical results are in good agreement with numerical simulations. This work can provide theoretical support for wave interaction phenomena in non-convex dispersive systems.

For $0<a<1$, we study the linearization $L_a$ about a smooth odd self-similar collapse profile $(\Omega,c_l)$ of the generalized Constantin-Lax-Majda equation, where $c_l$ is the focusing exponent. On the origin-$H^2$ realization $X=\{\varphi\in L^2(0,\infty):\varphi(0)=0,\varphi''\in L^2(0,\infty)\}$, under explicit transport, regularity, weighted-derivative, and far-field hypotheses, the first three Edmunds-Evans essential spectra are exactly the union of two possibly coincident vertical lines: $\operatorname{Re}\lambda=F=-1+c_l/2$ at infinity and $\operatorname{Re}\lambda=O=-\tilde c/2$ at the origin, where $\tilde c=c_l+a(H\Omega)(0)$ and $H$ is the Hilbert transform. Off these lines, $L_a-\lambda$ is Fredholm, with index zero on the exterior components, $+1$ in the open strip when $F<O$, and $-1$ when $O<F$. The fourth and fifth Edmunds-Evans spectra and the Browder essential spectrum equal the closed strip. Every point of the open strip with index $+1$ is an eigenvalue. Every fixed point of the Huang-Qin-Wang-Wei positive-advection construction with $0<a<400/(848-9\pi^2)$, after normalization, satisfies the full hypothesis package. For all sufficiently small positive $a$, every such fixed point has $F<O$. Thus actual collapse profiles realize a nonempty Browder band. An additional kernel-nondegeneracy condition (ND) yields exact individual kernel and cokernel dimensions. We prove it outside an explicit vertical energy slab and, at sufficiently high frequency, on every closed vertical substrip of the open band separated from the far-field line. Failures in the open band are locally finite and can accumulate at high frequency only toward that line.

We study quantum potentials containing Dirac delta function (DDF) within its supersymmetric (isospectral) construction via a discontinuous superpotential, instead of being simply an added part. This construction comprises of two independently solvable sectors joined at the singular point by matching conditions imposed by the corresponding self-adjoint extension. Needing careful regularization, these boundary conditions impose independent algebraic constraints on each of the eigenstates, that restrict and truncate the spectrum. The number of surviving bound states, which are required to be parity-even, is thus fixed by the available system parameters rather than by the usual quantization rule. The specific cases of the Harmonic oscillator and the Rosen-Morse potential isospectrally infused with the DDF demonstrate such highly restrictive spectra, while the similar combination of the DDF with a Calogero-type singularity turns out to be largely incompatible as the regularization breaks down. Therefore, localized singularities are capable of controlling and engineering the discrete spectrum of quantum systems, with possible use in myriads of physical systems with very localized interactions.

We prove, under some assumptions, the uniqueness of universal quantum dimension formulae. We first show that the quantum dimensions of classical algebras have a specific form that can be represented universally, in Vogel's sense, as the product/ratio of q-dimension factors with arguments $a\alpha+b\beta+c\gamma$ where $c=0, 1, 2$ and $a, b$ are integers. On this basis, we derive simple uniqueness criteria, according to which all known universal quantum dimensions are unique.

We present a novel approach to computing the Nijenhuis torsion and Frölicher-Nijenhuis brackets of recursion operators for symmetries, based on their full-fledged forms introduced by Jahnová and Vojčák (2024). In contrast to the conventional approach, which represents recursion operators as maps between shadows of nonlocal symmetries within a given covering, our method allows the Nijenhuis torsion to be computed directly from its defining formula, without requiring any additional mathematical constructions. The same framework can also be used to compute the Frölicher-Nijenhuis bracket of recursion operators and to verify their compatibility directly in their full-fledged forms. The procedure is illustrated by several examples of full-fledged recursion operators for symmetries of differential equations in four independent variables, including a new recursion operator for the four-dimensional universal hierarchy equation that, to the best of our knowledge, has not previously appeared in the literature. A notable feature of this new full-fledged recursion operator is that its shadow component does not appear to admit a reasonable representation in any of the standard conventional forms. Nevertheless, our approach allows us to prove directly that the Nijenhuis torsion of this operator vanishes. For the examples considered, the corresponding Frölicher-Nijenhuis brackets also vanish, confirming the compatibility of the respective pairs of recursion operators. This demonstrates that full-fledged forms provide an effective framework for studying the hereditary and compatibility properties of recursion operators, including highly nonlocal and/or multidimensional cases that are difficult to handle by existing methods.

This study presents a theoretical investigation of generalized solute dispersion in magnetohydrodynamic multiphase tube flow with porous layers. A two-fluid analytical model is developed for applications in biofluid and environmental fluid dynamics. The model comprises a micropolar (non-Newtonian) fluid core representing the rotational behaviour of red blood cells and a Newtonian plasma periphery embedded with Brinkman and Darcy porous structures, corresponding to the glycocalyx and endothelial layers with distinct permeability characteristics. A transverse magnetic field is incorporated to investigate how magnetic-field-induced modifications of the carrier flow influence solute localisation, with potential relevance to magnetic nanoparticle-mediated drug delivery. Using the generalised dispersion framework of Sankarasubramanian & Gill, analytical solutions are derived to investigate how the coupled axial velocity field and associated microrotational dynamics influence solute transport. The analytical predictions are independently validated through Brownian dynamics simulations, demonstrating excellent agreement for the temporal evolution of the zeroth and first transport moments. The results reveal the previously unexplored influence of microrotational dynamics on solute concentration, convection coefficients and effective dispersion, providing new insights into the coupled roles of translational and rotational fluid motion in biofluid transport. This work bridges an important gap in the literature and establishes a generalized theoretical framework linking magnetic fields, micropolar fluids and porous arterial structures for biofluid transport, targeted drug delivery and clinical engineering applications.

We determine the exact entanglement structure of the conditional state obtained by measuring $m$ corresponding rungs ($m\leq N/2$) in two identical copies of an arbitrary half-filled free fermion Gaussian state and post-selecting the same normalized single-fermion state ($|\psi\rangle = u|10\rangle+v|01\rangle$, where 0 and 1 denote the fermion occupancy on each sites) on each rung. For $m<N/2$, the conditional wavefunction generally depends on the initial state. Nevertheless, whenever the selected outcome has nonzero probability, the state on the unmeasured sites remains Gaussian and factorizes exactly into $N-m$ different orthogonal modes including $m$ inter-copy entangling modes and $N-2m$ spectator modes localized in one copy. Consequently, the entanglement entropy between the unmeasured parts of the two copies is $S=m h_2(|v|^2)$, independent of the initial state, where $h_2(x)=-x\ln x-(1-x)\ln(1-x)$. The success probability is given by $P_m=\det C_L\det(I_m-C_L)=\det(C_{LR}C_{RL})$, determined solely by the initial correlations and independent of ($u$,$v$). Equal-weight Bell post-selection serves as a special case that achieves the maximal entanglement swapping.

To test the potential of so-called quantum physics-informed neural network (QPINN) technology for solving evolutionary partial differential equations, we consider the problem of multiple wave propagation in a weakly nonlinear medium described by the Korteweg-de Vries equation with periodic boundary conditions. Although this problem is solvable analytically, high-genus solutions could represent a rather challenging testbed for numerical methods. Therefore, our idea is to test whether a QPINN is sufficiently mature for application in this context by providing a set of indicators of merit and performing several numerical experiments.